#8634519

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35109	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18,23,22)(19,24,26,21)(20,25,29,30)(27,31,32,28)(34,39)(36,40)(41,47)(44,45);(1,2)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,19)(18,24)(20,27)(21,22)(23,26)(25,31)(28,30)(29,32)(33,34,38,39)(35,40,43,36)(37,41,46,47)(42,45,48,44);(2,4)(7,9)(8,12)(15,16)(19,26)(21,24)(27,32)(28,31)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,45,47,44);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,36)(34,35)(37,45)(38,40)(39,43)(41,48)(42,47)(44,46);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,21)(18,19)(20,28)(22,26)(23,24)(25,27)(29,31)(30,32)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,47)(46,48);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,22,23,18)(19,21,26,24)(20,25,29,30)(27,31,32,28)(33,37,38,46)(34,41,39,47)(35,42,43,48)(36,44,40,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541715);